A new method to obtain explicit re-parameterization using piecewise rational linear functions is presented in this paper. Based on the outstanding performance of Mobius transformation on modifying pieces with monotonic parametric speed, we create a partition of the original curve, in which the parametric speed of each segment is of monotonic variation. The values of new parameters corresponding to the subdivision points are specified a priori as the ratio of its cumulative arc length and its total arc length. C1 continuity conditions are imposed to each segment, thus, with respect to the new parameters, the objective function is linear and admits a closed-form optimization. Analysis of examples shows that, our method brings a curve very close to the arc length parameterization under L2 norm but with fewer segments.